Suppose we choose a windowed sinusoidal FM signal for our mother chirplet.
Such a signal has a frequency that periodically
rises and falls
(much like the vibrato of musical instruments or the wail of a
police siren).

Within time-frequency space, conventional Doppler radar spectrograms treat
the motion of objects as though their velocities (Doppler
frequencies) were piecewise constant (constant over each of the
short time intervals), whereas the chirplet transform
attained a certain advantage by generalizing to a
piecewise constant acceleration model.

Originally, we had further extended the linear FM chirplet bases to
piecewise quadratic, and piecewise
cubic FM - piecewise polynomial approximations to the time-frequency
evolution of Doppler returns.
However, looking more closely at the underlying physics of floating objects,
which was our main motivation that led to our
discovery of the CCT, we observed a somewhat sinusoidal evolution
of the Doppler signals.
If you have ever watched a cork bobbing up and
down at the seaside, you would notice that it moves around in a circle
with a distinct periodicity.
It moves up and down, but it also moves
horizontally. Looking out at a target with a radar, for example,
we see the horizontal component of motion
(which is essentially a scaled version of the Hilbert transform of the vertical
movement).
This
sinusoidal
horizontal movement results in a sinusoidally varying frequency in the
Doppler return.

We wish to end up with the instantaneous frequency
of the basis function being given by:

f = (2f_mt+p)+f_c

where is the center (carrier) frequency,
p (which varies on the interval from 0 to )
is the relative position of one of the peak epochs in frequency,
with respect to the origin, and
is the modulation frequency.
If we are analyzing a discrete signal, s[nT],
we also note that must be less
than 1/2, otherwise the frequency modulation
is not bounded by the Nyquist limit.

Integrating to get the phase, we get:

= sin(2f_m t +p)f_m + 2f_ct

which gives us the family of chirplets defined by:

g_f_m,,f_c
= A e^j(sin(2f_m t +p)f_m) + j2f_ct

which may be appropriately windowed, such as with a Gaussian, as was done
in (3).

In Fig. 13, we show
four examples taken from a family of chirplets that were
derived from a warbling mother chirplet.
We show them in both the time-domain, and the TF domain,
annotated in terms of the pendulum model described now.

Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY

Pendulums swinging to and fro in front of a radar (assume the amplitude of the
swing is small compared to the length of the string)
produce a signal which is very similar to that produced by
radar returns from floating objects.
Suppose the velocity of a pendulum, as a function of time, is given
by:

A pendulum with a long string,
swinging with large amplitude in front of the radar will produce
a time series which,
will have most of its energy in the upper left hand portion of the
space (low and high ).
A density plot of the transform, computed from the time series will show
a strong peak in the upper left region,
with the peak located at the coordinates
corresponding to the particular
frequency of swinging () and amplitude .
A pendulum with a small swing, and a short length, will appear as an
energy concentration in the lower right corner of the pendulum parameter space.

we show the STFT computed from an actual radar return from a pendulum.

Using the warbling mother chirplet, we also computed the ``dilation-dilation''
() plane of the chirplet transform (Fig. 16)
for the pendulum data.

Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY

The members of the chirplet family given by (24)
may be regarded as being related to each other by affine coordinate
transformations in the time-frequency plane if we use
the rather abstract notion of instantaneous frequency.
Consider four functions
from (24), denoted A, B, C, and D,
corresponding to the four signals depicted
in Fig. 13 within the
plane formed by time and instantaneous frequency.
While there is no way to actually calculate this plane, we can
consider these functions as being defined by sinusoids in this plane.
Then A is a frequency-dilated version of C, and B is a frequency-dilated
version of D. While there is no practical means of dilating frequency
without contracting
time,
or vice-versa, we may denote the abstract
operator:

C_0,0,0,0,0,_t _f \: g(t)

as an operator that would magnify the time-frequency distribution of g(t).

When we write, for example,

C_0,0,0,0,0,2 \; g(t),

we mean
to replace g(t) with another function that occupies twice the area
in the TF plane. In general, such a function probably does not exist.
We noted, in the case of the prolate chirplet family, that we could,
however, vary the time-bandwidth product of the tiling by replacing the family
of mother chirplets with a new family that had a different value of NW.
By similar reasoning,
within the context of the warbling chirplet we
interpret (28) to mean ``replace g(t) with a new
sinusoidal-FM function that has times the modulation index
and times the modulation frequency'', so that, we obtain an
equal dilation by along each of the time and instantaneous
frequency axes. The result is a dilation of both the time and
instantaneous frequency axes by a factor of .
The law of composition, identity, and inverse,
within this six-parameter ``group''
is given by the usual two-dimensional affine group lawartin.

Therefore, we may
write the warbling chirplet transform in terms of
the six
affine coordinate transformations in the TF plane:

S_f_m,,f_c =

C_0,f_c,1f_m ,0,0,f_m \:
g(t)
|
s(t)

and refer to the subspace (Fig. 16) defined along the
and axes as the ``dilation-dilation'' plane, or the
plane.